Intersection pairing on an oriented surface ($\omega :H_1({\Sigma }_g;\mathbb{Z}{)}^2\to \mathbb{Z}$)

Represent one-dimensional homology classes on a closed oriented topological surface by transverse oriented curves and sum their signed crossing points. This gives a bilinear form that is skew-symmetric. A symplectic handle basis has matrix $\left(\begin{array}{cc}0& I_g\\ -I_g& 0\end{array}\right)$, which is unimodular. In particular, a class detected with intersection $1$ or $-1$ by another integral class is a primitive homology class: the detecting homomorphism splits its rank-one subgroup.